1. 1 Stochastic Maintenance Scheduling Problem G. Fleury, P. Lacomme, M. Sevaux Laboratoire d’Informatique Clermont-Ferrand UMR 6158 Laboratoire de Mathématiques Clermont-Ferrand UMR 6620 LAMIHValenciennes UMR 8530

2. 2 Plan  Problem statement  Assumptions and objective  Genetic Algorithm template  Computational experiments  Future research

3. 3 Problem statement 10 000 elementary tasks 8 majors operations for each coach 64 aggregated tasks for one TGV Objective: minimize the total duration

4. 4 Physical description (1)

5. 5 Physical description (2)  CTAx (caisses TGV et Automoteur): Dis-assembling tasks Re-assembling tasks Works insided coaches  IP (industries privées): Sand blasting by external companies  TSCx (tôlerie, stucture de caisse): Handling the tollery Renovation of external parts of coaches

6. 6 Logical description (1) jobs sequence of treatment

7. 7 Logical description (2)

8. 8 A stochastic problem (1)  Processing time of jobs are submitted to variations  Robust solutions are required to avoid periodic computation of new schedule  Minimization of the makespan is also required

9. 9 Random events modelization   : extra delay  pp : probability of random events occurrences

10. 10 A template for stochastic problem (1)

11. 11 A template for stochastic problem (2)  Optimization phase:  Searching process based of statistic performances of solutions  Robustness evaluation of solutions  Replications  Average cost of solution  Standard deviation of solutions

12. 12 Genetic Algorithm template (1) Construct a random initial set of solutions Repeat Select P1 and P2 based on the inverse function of the fitness rank distribution Select P1 and P2 based on the inverse function of the fitness rank distribution Apply XOver operator Apply XOver operator Evaluate C With probability P then Mutate C (swap two random points p and q) Mutate C (swap two random points p and q) Until (a maximal number of iterations is reached).  See (Sevaux and Le Quéré, 2003)

13. 13 Genetic Algorithm template (2) One chromosome is: –Ordered set of jobs –Evaluation of the average cost –Evaluation of the standard deviation cost

14. 14 Robust Approach  Principles  Compute which is a evaluation of the average cost over n replications  Compute which is evaluation of the standard deviation over n replications which is evaluation of the standard deviation over n replications  Problems Very costly for a computational point of view

15. 15 Stochastic Approach (1)  Replace statistical evaluation by mathematical evaluation  Based on shortest path computed in the disjunctive graph

16. 16 Stochastic Approach (2)  Tasks duration  with    Y binomial law

17. 17 Stochastic Approach (3)  So  Average :  Standard deviation :  Finally:

18. 18 Results for the robust approach (Sevaux and Le Quéré, 2003)

19. 19 Results for the robust approach Results with mathematical evaluation of criteria

20. 20 Concluding remarks (1)  Stochastic maintenance problem  Two approaches: A robust approach A stochastic approach  Both approaches provides robust solutions

21. 21 Concluding remarks (2)  Robust Approach High quality solutions Post analysis provide results very closed to the evaluations Time consuming  Stochastic Approach Satisfactory evaluation of soluitons Very short computational time

22. 22 Future Research  Improve mathematical analysis  Take into account all shortest paths  Improve modelization of the problem  modelize random variations